Dirichlet series
| L(χ,s) = 1 | + (0.638 − 0.769i)2-s + (−0.117 − 0.993i)3-s + (−0.184 − 0.982i)4-s + (−0.713 + 0.701i)5-s + (−0.839 − 0.543i)6-s + (−0.440 + 0.897i)7-s + (−0.874 − 0.485i)8-s + (−0.972 + 0.234i)9-s + (0.0843 + 0.996i)10-s + (0.0168 − 0.999i)11-s + (−0.954 + 0.299i)12-s + (−0.874 + 0.485i)13-s + (0.409 + 0.912i)14-s + (0.780 + 0.625i)15-s + (−0.931 + 0.363i)16-s + (−0.758 + 0.651i)17-s + ⋯ |
| L(s,χ) = 1 | + (0.638 − 0.769i)2-s + (−0.117 − 0.993i)3-s + (−0.184 − 0.982i)4-s + (−0.713 + 0.701i)5-s + (−0.839 − 0.543i)6-s + (−0.440 + 0.897i)7-s + (−0.874 − 0.485i)8-s + (−0.972 + 0.234i)9-s + (0.0843 + 0.996i)10-s + (0.0168 − 0.999i)11-s + (−0.954 + 0.299i)12-s + (−0.874 + 0.485i)13-s + (0.409 + 0.912i)14-s + (0.780 + 0.625i)15-s + (−0.931 + 0.363i)16-s + (−0.758 + 0.651i)17-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(\chi,s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr
=\mathstrut & (0.287 + 0.957i)\, \Lambda(\overline{\chi},1-s)
\end{aligned}
\]
\[\begin{aligned}
\Lambda(s,\chi)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr
=\mathstrut & (0.287 + 0.957i)\, \Lambda(1-s,\overline{\chi})
\end{aligned}
\]
Invariants
| \( d \) | = | \(1\) |
| \( N \) | = | \(373\) |
| \( \varepsilon \) | = | $0.287 + 0.957i$ |
| motivic weight | = | \(0\) |
| character | : | $\chi_{373} (124, \cdot )$ |
| Sato-Tate | : | $\mu(93)$ |
| primitive | : | yes |
| self-dual | : | no |
| analytic rank | = | 0 |
| Selberg data | = | $(1,\ 373,\ (0:\ ),\ 0.287 + 0.957i)$ |
| $L(\chi,\frac{1}{2})$ | $\approx$ | $0.1401796540 + 0.1042736172i$ |
| $L(\frac12,\chi)$ | $\approx$ | $0.1401796540 + 0.1042736172i$ |
| $L(\chi,1)$ | $\approx$ | 0.6747782222 - 0.3980796365i |
| $L(1,\chi)$ | $\approx$ | 0.6747782222 - 0.3980796365i |
Euler product
\[\begin{aligned}
L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]
\[\begin{aligned}
L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}
\end{aligned}\]